Bisection hardly ever converges linearly (Q1347031)

A real number is called diadic if it is the sum of finitely many integral powers of two. The following theorem is proved: Let \(f: {\mathcal D}\mapsto\mathbb{R}\) be defined on a set containing all diadic numbers in \([0,1]\) and \(f(0)< 0< f(1)\). From the starting values \(a_ 0= 0\), \(b_ 0= 1\) the bisection method converges linearly to its limit \(r\) if, and only if, either \(r\) is a diadic point of discontinuity where \(f(r)\neq 0\), or \(r= (2a_ n+ b_ n)/3\) for some positive integer \(n\). For all other limit points the method still converges but the order of convergence remains undefined. Note that for bisection to converge it suffices that \(f\) is defined at all diadic numbers in \([0,1]\) and that it need neither be continuous nor measurable.